Edexcel S2 (Statistics 2) 2016 January

The manager of a clothing shop wishes to investigate how satisfied customers are with the quality of service they receive. A database of the shop's customers is used as a sampling frame for this investigation.

1. Identify one potential problem with this sampling frame. [1]

Customers are asked to complete a survey about the quality of service they receive. Past information shows that 35\% of customers complete the survey. A random sample of 20 customers is taken.

1. Write down a suitable distribution to model the number of customers in this sample that complete the survey. [2]
2. Find the probability that more than half of the customers in the sample complete the survey. [2]

The continuous random variable \(X\) is uniformly distributed over the interval \([a, b]\) Given that \(\mathrm{P}(3 < X < 5) = \frac{1}{8}\) and \(\mathrm{E}(X) = 4\)

1. find the value of \(a\) and the value of \(b\) [3]
2. find the value of the constant, \(c\), such that \(\mathrm{E}(cX - 2) = 0\) [2]
3. find the exact value of \(\mathrm{E}(X^2)\) [3]
4. find \(\mathrm{P}(2X - b > a)\) [2]

Left-handed people make up 10\% of a population. A random sample of 60 people is taken from this population. The discrete random variable \(Y\) represents the number of left-handed people in the sample.

1. 1. Write down an expression for the exact value of \(\mathrm{P}(Y \leq 1)\)
   2. Evaluate your expression, giving your answer to 3 significant figures. [3]
2. Using a Poisson approximation, estimate \(\mathrm{P}(Y \leq 1)\) [2]
3. Using a normal approximation, estimate \(\mathrm{P}(Y \leq 1)\) [5]
4. Give a reason why the Poisson approximation is a more suitable estimate of \(\mathrm{P}(Y \leq 1)\) [1]

A continuous random variable \(X\) has cumulative distribution function $$\mathrm{F}(x) = \begin{cases} 0 & x < 0 \\ \frac{1}{4}x & 0 \leq x \leq 1 \\ \frac{1}{20}x^4 + \frac{1}{5} & 1 < x \leq d \\ 1 & x > d \end{cases}$$

1. Show that \(d = 2\) [2]
2. Find \(\mathrm{P}(X < 1.5)\) [2]
3. Write down the value of the lower quartile of \(X\) [1]
4. Find the median of \(X\) [3]
5. Find, to 3 significant figures, the value of \(k\) such that \(\mathrm{P}(X > 1.9) = \mathrm{P}(X < k)\) [4]

The number of eruptions of a volcano in a 10 year period is modelled by a Poisson distribution with mean 1

1. Find the probability that this volcano erupts at least once in each of 2 randomly selected 10 year periods. [2]
2. Find the probability that this volcano does not erupt in a randomly selected 20 year period. [2]

The probability that this volcano erupts exactly 4 times in a randomly selected \(w\) year period is 0.0443 to 3 significant figures.

1. Use the tables to find the value of \(w\) [3]

A scientist claims that the mean number of eruptions of this volcano in a 10 year period is more than 1 She selects a 100 year period at random in order to test her claim.

1. State the null hypothesis for this test. [1]
2. Determine the critical region for the test at the 5\% level of significance. [2]

A continuous random variable \(X\) has probability density function $$\mathrm{f}(x) = \begin{cases} ax^2 + bx & 1 \leq x \leq 7 \\ 0 & \text{otherwise} \end{cases}$$ where \(a\) and \(b\) are constants.

1. Show that \(114a + 24b = 1\) [4]

Given that \(a = \frac{1}{90}\)

1. use algebraic integration to find \(\mathrm{E}(X)\) [4]
2. find the cumulative distribution function of \(X\), specifying it for all values of \(x\) [3]
3. find \(\mathrm{P}(X > \mathrm{E}(X))\) [2]
4. use your answer to part (d) to describe the skewness of the distribution. [2]

A fisherman is known to catch fish at a mean rate of 4 per hour. The number of fish caught by the fisherman in an hour follows a Poisson distribution. The fisherman takes 5 fishing trips each lasting 1 hour.

1. Find the probability that this fisherman catches at least 6 fish on exactly 3 of these trips. [6]

The fisherman buys some new equipment and wants to test whether or not there is a change in the mean number of fish caught per hour. Given that the fisherman caught 14 fish in a 2 hour period using the new equipment,

1. carry out the test at the 5\% level of significance. State your hypotheses clearly. [6]